Baclone transformation for a system of partial differential equations of the third order

Abstract

Clerain's ideas have been developed and the construction of differential relations of two nonlinear systems in partial derivatives has been realized. An analysis of a given system of equations of the third order is carried out. The general structure of the Backlund transformations is defined in the form of four differential equations. Differential couplings are defined so that they give the opportunity to get both systems. Proceeding from the fact that the initial system is linear in terms of the highest derivative of a spatial variable and contains only the irst-order derivative of the time variable, it was possible in the Baklund transformations to explicitly select the second derivatives with respect to the spatial variable and to clarify the relationship between the lower derivatives. It turned out that such connections are determined ambiguously. There are two honest cases that allow you to make a transition from one system to another.